Terahertz emission by diffusion of carriers and metal-mask dipole inhibition of radiation

M. E. Barnes; D. McBryde; G. J. Daniell; G. Whitworth; A. L. Chung; A. H. Quarterman; K. G. Wilcox; A. Brewer; H. E. Beere; D. A. Ritchie; V. Apostolopoulos

doi:10.1364/OE.20.008898

Terahertz emission by diffusion of carriers and metal-mask dipole inhibition of radiation

M. E. Barnes, D. McBryde, G. J. Daniell, G. Whitworth, A. L. Chung, A. H. Quarterman, K. G. Wilcox, A. Brewer, H. E. Beere, D. A. Ritchie, and V. Apostolopoulos

Optics Express
Vol. 20,
Issue 8,
pp. 8898-8906
(2012)
•https://doi.org/10.1364/OE.20.008898

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M. E. Barnes, D. McBryde, G. J. Daniell, G. Whitworth, A. L. Chung, A. H. Quarterman, K. G. Wilcox, A. Brewer, H. E. Beere, D. A. Ritchie, and V. Apostolopoulos, "Terahertz emission by diffusion of carriers and metal-mask dipole inhibition of radiation," Opt. Express 20, 8898-8906 (2012)

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Related Topics
Table of Contents Category
- Ultrafast Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Spectroscopy, teraherz (300.6495)
- Ultrafast processes in condensed matter, including semiconductors (320.7130)

About this Article
History
- Original Manuscript: February 14, 2012
- Revised Manuscript: March 27, 2012
- Manuscript Accepted: March 27, 2012
- Published: April 2, 2012

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Open Access

Abstract

Terahertz (THz) radiation can be generated by ultrafast photo-excitation of carriers in a semiconductor partly masked by a gold surface. A simulation of the effect taking into account the diffusion of carriers and the electric field shows that the total net current is approximately zero and cannot account for the THz radiation. Finite element modelling and analytic calculations indicate that the THz emission arises because the metal inhibits the radiation from part of the dipole population, thus creating an asymmetry and therefore a net current. Experimental investigations confirm the simulations and show that metal-mask dipole inhibition can be used to create THz emitters.

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References

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|
Year
|
Author
|
Publication

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]
M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]
K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]
P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]
M. B. Johnston, D. M. Whittaker, A. Dowd, A. G. Davies, E. H. Linfield, X. Li, and D. A. Ritchie, “Generation of high-power terahertz pulses in a prism,” Opt. Lett. 27, 1935–1937 (2002).
[Crossref]
G. Klatt, B. Surrer, D. Stephan, O. Schubert, M. Fischer, J. Faist, A. Leitenstorfer, R. Huber, and T. Dekorsy, “Photo-Dember terahertz emitter excited with an Er:fiber laser,” Appl. Phys. Lett. 98, 021114 (2011).
[Crossref]
G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010).
[Crossref] [PubMed]
G. Klatt, F. Hilser, W. Chao, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Intense terahertz generation based on the photo-Dember effect,” in Proceedings of OSA/CLEO/QELS 2010 Paper CMJJ2 (2010).
D. McBryde, M. E. Barnes, A. L. Chung, Z. Mihoubi, G. J. Daniell, A. H. Quarterman, K. G. Wilcox, H. E. Beere, D. A. Ritchie, A. C. Tropper, and V. Apostolopoulos, “Simulation of metallic nanostructures for emission of THz radiation using the lateral photo-Dember effect,” in Proceedings of The 36th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz), pp.1–2 (2011).
[Crossref]
D. McBryde, M. E. Barnes, A. L. Chung, Z. Mihoubi, G. J. Daniell, A. H. Quarterman, K. G. Wilcox, H. E. Beere, D. A. Ritchie, A. C. Tropper, and V. Apostolopoulos, “Simulation of metallic nanostructures for emission of THz radiation using the lateral photo-Dember effect,” arXiv:1202.1459v1, (2012), http://arxiv.org/abs/1202.1459v1 .
K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Luminol. 1, 693–701 (1970).
[Crossref]
T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]
J. S. Blakemore, “Semiconducting and other major properties of gallium-arsenide,” J. Appl. Phys. 53, R123–R181 (1982).
[Crossref]
I. S. Gregory, W. R. Tribe, C. Baker, B. E. Cole, M. J. Evans, L. Spencer, M. Pepper, and M. Missous, “Continuous-wave terahertz system with a 60 dB dynamic range,” Appl. Phys. Lett. 86, 204104 (2005).
[Crossref]
I. S. Gregory, “The development of a continuous-wave terahertz imaging system,” Ph.D. thesis, University of Cambridge (2004).
C. Baker, “Development of semiconductor materials for terahertz photoconductive antennas,” Ph.D. thesis, University of Cambridge (2004).
E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]
H. Yasuda and I. Hosako, “Measurement of terahertz refractive index of metal with terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 47, 1632–1634 (2008).
[Crossref]
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with formulas, graphs, and mathematical tables (Dover Publications, New York, 1972).

2011 (3)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]

G. Klatt, B. Surrer, D. Stephan, O. Schubert, M. Fischer, J. Faist, A. Leitenstorfer, R. Huber, and T. Dekorsy, “Photo-Dember terahertz emitter excited with an Er:fiber laser,” Appl. Phys. Lett. 98, 021114 (2011).
[Crossref]

D. McBryde, M. E. Barnes, A. L. Chung, Z. Mihoubi, G. J. Daniell, A. H. Quarterman, K. G. Wilcox, H. E. Beere, D. A. Ritchie, A. C. Tropper, and V. Apostolopoulos, “Simulation of metallic nanostructures for emission of THz radiation using the lateral photo-Dember effect,” in Proceedings of The 36th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz), pp.1–2 (2011).
[Crossref]

2010 (2)

G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010).
[Crossref] [PubMed]

G. Klatt, F. Hilser, W. Chao, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Intense terahertz generation based on the photo-Dember effect,” in Proceedings of OSA/CLEO/QELS 2010 Paper CMJJ2 (2010).

2008 (1)

H. Yasuda and I. Hosako, “Measurement of terahertz refractive index of metal with terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 47, 1632–1634 (2008).
[Crossref]

2006 (1)

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

2005 (1)

I. S. Gregory, W. R. Tribe, C. Baker, B. E. Cole, M. J. Evans, L. Spencer, M. Pepper, and M. Missous, “Continuous-wave terahertz system with a 60 dB dynamic range,” Appl. Phys. Lett. 86, 204104 (2005).
[Crossref]

2004 (2)

I. S. Gregory, “The development of a continuous-wave terahertz imaging system,” Ph.D. thesis, University of Cambridge (2004).

C. Baker, “Development of semiconductor materials for terahertz photoconductive antennas,” Ph.D. thesis, University of Cambridge (2004).

2002 (3)

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

M. B. Johnston, D. M. Whittaker, A. Dowd, A. G. Davies, E. H. Linfield, X. Li, and D. A. Ritchie, “Generation of high-power terahertz pulses in a prism,” Opt. Lett. 27, 1935–1937 (2002).
[Crossref]

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

1997 (1)

T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]

1993 (1)

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

1982 (1)

J. S. Blakemore, “Semiconducting and other major properties of gallium-arsenide,” J. Appl. Phys. 53, R123–R181 (1982).
[Crossref]

1970 (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Luminol. 1, 693–701 (1970).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with formulas, graphs, and mathematical tables (Dover Publications, New York, 1972).

Apostolopoulos, V.

Baker, C.

C. Baker, “Development of semiconductor materials for terahertz photoconductive antennas,” Ph.D. thesis, University of Cambridge (2004).

Barnes, M. E.

Bartels, A.

Bastian, G.

Beck, M.

Beere, H. E.

Blakemore, J. S.

J. S. Blakemore, “Semiconducting and other major properties of gallium-arsenide,” J. Appl. Phys. 53, R123–R181 (1982).
[Crossref]

Chao, W.

Chung, A. L.

Cole, B. E.

Cooke, D. G.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]

Corchia, A.

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

Daniell, G. J.

Davies, A. G.

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

Dekorsy, T.

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

Doi, T.

T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]

Dowd, A.

Drexhage, K.

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Luminol. 1, 693–701 (1970).
[Crossref]

Evans, M. J.

Faist, J.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).

Fischer, M.

Gebs, R.

Gregory, I. S.

I. S. Gregory, “The development of a continuous-wave terahertz imaging system,” Ph.D. thesis, University of Cambridge (2004).

Gu, P.

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

Hilser, F.

Hosako, I.

H. Yasuda and I. Hosako, “Measurement of terahertz refractive index of metal with terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 47, 1632–1634 (2008).
[Crossref]

Huber, R.

Huska, K.

Jepsen, P. U.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]

Johnston, M. B.

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

Klatt, G.

Koch, M.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]

Kono, S.

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

Kurz, H.

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

Kütt, W.

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

Leitenstorfer, A.

Lemmer, U.

Li, X.

Linfield, E.

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

Linfield, E. H.

Liu, K.

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

McBryde, D.

Mihoubi, Z.

Missous, M.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

Pepper, M.

Pfeifer, T.

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

Qiao, W.

Quarterman, A. H.

Ritchie, D. A.

Sakai, K.

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

Schubert, O.

Spencer, L.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with formulas, graphs, and mathematical tables (Dover Publications, New York, 1972).

Stephan, D.

Surrer, B.

Tani, M.

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

Tanimura, Y.

T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]

Toyoda, K.

T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]

Tribe, W. R.

Tropper, A. C.

Whittaker, D.

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

Whittaker, D. M.

Wilcox, K. G.

Xu, J. Z.

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

Yasuda, H.

H. Yasuda and I. Hosako, “Measurement of terahertz refractive index of metal with terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 47, 1632–1634 (2008).
[Crossref]

Yuan, T.

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

Zhang, X. C.

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

Appl. Opt. (1)

T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. 36, 7157–7161 (1997).
[Crossref]

Appl. Phys. Lett. (2)

J. Appl. Phys. (2)

P. Gu, M. Tani, S. Kono, K. Sakai, and X. C. Zhang, “Study of terahertz radiation from InAs and InSb,” J. Appl. Phys. 91, 5533–5537 (2002).
[Crossref]

J. S. Blakemore, “Semiconducting and other major properties of gallium-arsenide,” J. Appl. Phys. 53, R123–R181 (1982).
[Crossref]

J. Luminol. (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Luminol. 1, 693–701 (1970).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. Yasuda and I. Hosako, “Measurement of terahertz refractive index of metal with terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 47, 1632–1634 (2008).
[Crossref]

Laser Photon. Rev. (1)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (3)

M. B. Johnston, D. Whittaker, A. Corchia, A. G. Davies, and E. Linfield, “Simulation of terahertz generation at semiconductor surfaces,” Phys. Rev. B 65, 165301 (2002).
[Crossref]

K. Liu, J. Z. Xu, T. Yuan, and X. C. Zhang, “Terahertz radiation from InAs induced by carrier diffusion and drift,” Phys. Rev. B 73, 155330 (2006).
[Crossref]

T. Dekorsy, T. Pfeifer, W. Kütt, and H. Kurz, “Subpicosecond carrier transport in GaAs surface-space-charge fields,” Phys. Rev. B 47, 3842–3849 (1993).
[Crossref]

Proceedings of OSA/CLEO/QELS 2010 (1)

Proceedings of The 36th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz) (1)

Other (6)

D. McBryde, M. E. Barnes, A. L. Chung, Z. Mihoubi, G. J. Daniell, A. H. Quarterman, K. G. Wilcox, H. E. Beere, D. A. Ritchie, A. C. Tropper, and V. Apostolopoulos, “Simulation of metallic nanostructures for emission of THz radiation using the lateral photo-Dember effect,” arXiv:1202.1459v1, (2012), http://arxiv.org/abs/1202.1459v1 .

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with formulas, graphs, and mathematical tables (Dover Publications, New York, 1972).

I. S. Gregory, “The development of a continuous-wave terahertz imaging system,” Ph.D. thesis, University of Cambridge (2004).

C. Baker, “Development of semiconductor materials for terahertz photoconductive antennas,” Ph.D. thesis, University of Cambridge (2004).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

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Fig. 1 Illustration of the LPD effect showing how the laser beam is partly masked by a metallic region.

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Fig. 2 SEM of a 2D multiplex emitter based on the concepts given in [7] which failed to produce any measurable THz emission.

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Fig. 3 (a), (b), (c), representation of the electron density, hole density and current density, respectively. Each curve is at 20 ps intervals within a 80 ps time frame; progression of time corresponds to a fall of the peak concentration or peak current density.

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Fig. 4 (a) and (b) show finite element modelling of a simplified LPD emitter represented by two oscillating currents in anti-phase, (a) is modelled without gold masking and (b) with gold masking.(c) Shows the mechanism of radiation in (b) where the dipoles under the metal are suppressed by the reflection of the metal surface. (d) From the numerical solution of Eq. (1) with a photoexcitation term describing an 1 ps Gaussian optical pulse centered at 5 ps the graph shows the dJ/dt for the total carrier population without taking into account the dipole suppression and the dJ/dt calculated only for the carrier population which is not under metal and therefore free to radiate. (e) Shows a plot of Eq. (2) corresponding to the electric field of a dipole radiating at 2 THz at a distance of 1 μm below the metal.

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Fig. 5 Diagram showing the placement of the laser beam on (a) the bowtie PC emitter and (b) the LPD emitter. (c) Shows the current recorded in the time domain for a PC antenna used as a PC and a LPD emitter. (d) shows THz emission at two opposite boundaries to observe the predicted sign change. (e) shows the Fourier transform of (a) and (b) demonstrating comparable bandwidth between the PC and LPD emitter. The experiment was performed in ambient atmosphere.

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Equations (2)

Equations on this page are rendered with MathJax. Learn more.

\frac{\partial n_{e}}{\partial t} = μ \frac{\partial}{\partial x} (n_{e} E) + D \frac{\partial^{2} n_{e}}{\partial x^{2}} - \frac{n_{e} n_{h}}{τ_{1}} - \frac{n_{e}}{τ_{2}}

\begin{array}{r} E = & \frac{1}{2 \sqrt{i π}} [e^{{iky}_{0}} Φ (\sqrt{k (r_{0} + y_{0})}) \\ - e^{- {iky}_{0}} Φ (\pm \sqrt{k (r_{0} - y_{0})})] \end{array}